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Panel data: characteristics (Panel data Multivariate (multiple emotions) Repeated (9*7 measurement occasions) Sources of dependency Between-person variability Within-person variability Between days Serial correlation within a day (co-occurrence of emotions within and across measurement occasions)  F5-D F5-D      &Modeling the dependency structure: Latent variables for between-person and -day variability observation- or parameter-driven process for the serial correlation within a day Approach we follow Discrete response and latent variables -> latent class models Transitions between states (classes) modelled as a Markov process (parameter-driven process) Known as latent transition or markov models #ZZZ'ZZZZZ#A   '%Exemplary model: latent markov model &&$%2 main assumptions: at a given measurement occasion t, the responses Yit1 ,& ,YitJ are independent, given latent state Zit latent state at measurement occasion t depends on past latent states through latent state at measurement occasion t-1 only %%L  ,WMarginal probability of a response pattern yi where summation is over ST terms 8+  W - 14 state model, ignoring dependencies between days22(14$Transition probabilities5%State probabilities over time 0"Statement of the problemEstimation (ML framework) involves computation of marginal probabilities: Brute force summation: ST terms (49=262144) More efficient algorithms exist -> probabilistic graphical modelsNd   MProbabilistic graphical modelskPrinciple Given: statistical model for set of (discrete) random variables Translate statistical model into graph Nodes correspond to random variables Conditional dependency relations are represented by edges -> visualization Apply transformations to the graph Transformed graph can be used to factorize joint probability distribution -> efficient computation  ZhZ_ZZmZZZ h_   3s3k1#>Construction of directed acyclic graph for latent markov model??$> : d-separation (Pearl,  86) in the DAG implies conditional independence relations in statistical model Zi2 d-separates Zi1 and Zi3 : Zi1 and Zi3 are conditionally independent, given Zi2 Zi1 d-separates Yi11& Yi1J: Yi11& Yi1J are conditionally independent, given Zi1 [Zi1 -> Zi2 <- Zi3 : no d-separation: given Zi2 , Zi1 and Zi3 are conditionally dependent in at least one distribution compatible with the DAG] iZ5Zi           &                %               S  Applying transformationsMoralization:  marry all non-adjacent parents Drop directions &55C Triangulation Add edges so that chordless cycles contain no more than three nodes Chordless: none other than successive pairs of nodes in the cycle are adjacent BOD"   ]D  (Junction tree Tree of cliques (clique = maximal subset of nodes that are all interconnected) Has running intersection property: each node that appears in any two cliques also appears in all cliques on the undirected path between these two cliques Separators: intersection of the linked cliques Hr"   *Factorizing the joint probability function++$* Given the junction tree representation, the joint probability function can be factorized as the product of clique marginals over separator marginals: Leads to efficient computational schemes Can be done automatically by applying algorithms from graph theory /ZZEZ"""("""C""  EM-algorithm (Lauritzen, 1995)E-step efficient computation of posterior probabilities of hidden variables based on junction tree factorization (Jensen, Lauritzen, & Olesen, 1990) M-step as usual For the latent markov model, this is the forward-backward algorithm (the snake bites its tail) j ` `/Application: a hierarchical latent markov model00/pLatent classes on signal- and day-level Graphical representation 2 classes for signal- and day-level rm '$(p:&Transition probabilitiesState probabilities over time  Multinomial regressionWHitherto: no restrictions on conditional probabilities explosion of number of parameters with increasing number of parents increasing number of states of hidden variables Restrictions can be incorporated by modeling the conditional probabilities under a multinomial logistic regression model E.g. incorporating only main effects of parents T`Ny1`Ny1W<'DResponse probabilities were restricted: Main effects of items Interaction between day-state and positive vs. negative emotions Interaction between signal-state and positive vs. negative emotions > states only differ wrt shift common to all Positive emotions Negative emotions Shifts for day- and signal-states are additive N(S(SDL1 >(Saturated model for transition probabilities Reduction from 57 to 32 parameters Restricted model is rejected (LR =352.4, df = 32, p<.001)Hn ?)Incorporation of covariatesCovariates can be incorporated as additional predictors in the multinomial regression models we now factorize Pr(x|covariates) instead of Pr(x) according to the junction tree Alternatively, incorporate covariates as nodes in the graph and factorize Pr(x, covariates) More complex graphs Not possible when dealing with continuous covariates for discrete (hidden) outcome variables Must when missing data for covariates qn A* Before: fixed time intervals were assumed between two signals <-> stratified random administration Test: time interval as covariate for transition probabilities between latent states at signal-level No effect (LR = 7, df = 4, p=.14; LR = 12.4, df = 8, p = .13)  Z   C+ Model testing Model testing Comparing nested models: likelihood-ratio test Not applicable when nested model on the boundary of the parameter space of the unrestricted model (e.g. testing 3-state vs. 4 state-model) Global goodness-of-fit: Sparse data, reference distribution of LR-test unknown AIC, BIC (but are also deviance-based ???) Alternative: model diagnostics Comparison of expected to observed log-odds ratios (averaged) 0?0?D,G-H.= Underestimation of dependencies between positive emotions dependencies between negative emotions Worst for item pairs belonging to same emotional category Latent markov model: better in explaining dependencies within a day Hierarchical latent markov model: better in explaining dependencies over days Not a surprise<ZNZZN=I/2Note on software BNT matlab toolbox (Murphy, 2004) for operations on graph (construction junction tree) Set of matlab functions for EM-algorithm Multinomial logistic nodes (or cumulative, or adjacent-categories) Incorporation of covariates Nodes can have same CPT s or different CPT s but sharing same set of parameters (e.g. when different values on covariates) parameter restrictions Continuous latent variables (gaussian quadrature or npml) M-step: Fischer scoring Information matrix: numerical differentiation of score function in MLE s For basic models: design matrix is constructed automatically For more advanced models: design matrix is defined by the user FPP P J0 Conclusion  TGraphical models Visualization Computational efficiency Conditional independence relations are derived automatically, regardless of numerical specifications Can also be applied to continuous variables and mixed sets of discrete and continuous variables (but& ) Posterior configurations, trajectories*ZZ*   0` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> (    6L " `}  B Click to edit Master title style!  04 " `  RClick to edit Master text styles Second level Third level Fourth level Fifth level!      0$ "^ `  >*  0̝ "^   @*  0ܡ "^ `  @*H  0޽h ? 3380___PPT10.zV\F Default Design} 0  $(  r  S yX> X r  S 3 `   X H  0޽h ? 33___PPT10i.zV\F+D=' W= @B +}  0 0 $(   r  S +D=' W= @B +  0 `Z(  r  S kZ `  ^  6j  S A K??"?M K^  6j  S A  L??"?c   LH  0޽h ? 33___PPT10i.-pV+D=' W= @B +4  0 p4(  r  S \ `}     BA{? 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